Theorem tpssd 32633

Description: Deduction version of tpssi : An unordered triple of elements of a class is a subset of that class. (Contributed by Thierry Arnoux, 2-Nov-2025.)

Hypotheses

Ref	Expression
tpssd.1	⊢ (𝜑 → 𝐴 ∈ 𝐷)
tpssd.2	⊢ (𝜑 → 𝐵 ∈ 𝐷)
tpssd.3	⊢ (𝜑 → 𝐶 ∈ 𝐷)

Assertion

Ref	Expression
tpssd	⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Proof of Theorem tpssd

Step	Hyp	Ref	Expression
1		tpssd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
2		tpssd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
3		tpssd.3	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
4		tpssi 4776	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
5	1, 2, 3, 4	syl3anc 1379	1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Syntax hints:   → wi 4   ∈ wcel 2119   ⊆ wss 3890  {ctp 4566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-un 3895  df-ss 3907  df-sn 4563  df-pr 4565  df-tp 4567

This theorem is referenced by:  constrlccllem  33944  constrcccllem  33945  cos9thpiminplylem2  33974